Half-integral weight modular forms and modular forms for Weil representations

MAT1200: Modular Forms of Half Integral Weight

On “good” Half-integral Weight Modular Forms

If k is a positive integer, let Sk(N) denote the space of cusp forms of weight k on Γ1(N), and let S k (N) denote the subspace of Sk(N) spanned by those forms having complex multiplication (see [Ri]). For a non-negative integer k and any positive integer N ≡ 0 (mod 4), let Mk+ 2 (N) (resp. Sk+ 2 (N)) denote the space of modular forms (resp. cusp forms) of half-integral weight k + 12 on Γ1(N). S...

Coefficients of Half-integral Weight Modular Forms

In this paper we study the distribution of the coefficients a(n) of half integral weight modular forms modulo odd integers M . As a consequence we obtain improvements of indivisibility results for the central critical values of quadratic twists of L-functions associated with integral weight newforms established in [O-S]. Moreover, we find a simple criterion for proving cases of Newman’s conject...

Fourier Coefficients of Half - Integral Weight Modular Forms modulo `

S. Chowla conjectured that every prime p has the property that there are infinitely many imaginary quadratic fields whose class number is not a multiple of p. Gauss’ genus theory guarantees the existence of infinitely many such fields when p = 2, and the work of Davenport and Heilbronn [D-H] suffices for the prime p = 3. In addition, the DavenportHeilbronn result demonstrates that a positive pr...

p-adic interpolation of half-integral weight modular forms

The p-adic interpolation of modular forms on congruence subgroups of SL2(Z) has been succesfully used in the past to interpolate values of L-series. In [12], Serre interpolated the values at negative integers of the ζ-series of a totally real number field (in fact of L-series of powers of the Teichmuller character) by interpolating Eisenstein series, which are holomorphic modular forms, and loo...